2,922 research outputs found
Slime mould computes planar shapes
Computing a polygon defining a set of planar points is a classical problem of
modern computational geometry. In laboratory experiments we demonstrate that a
concave hull, a connected alpha-shape without holes, of a finite planar set is
approximated by slime mould Physarum polycephalum. We represent planar points
with sources of long-distance attractants and short-distance repellents and
inoculate a piece of plasmodium outside the data set. The plasmodium moves
towards the data and envelops it by pronounced protoplasmic tubes
Finding Convex Hulls Using Quickhull on the GPU
We present a convex hull algorithm that is accelerated on commodity graphics
hardware. We analyze and identify the hurdles of writing a recursive divide and
conquer algorithm on the GPU and divise a framework for representing this class
of problems. Our framework transforms the recursive splitting step into a
permutation step that is well-suited for graphics hardware. Our convex hull
algorithm of choice is Quickhull. Our parallel Quickhull implementation (for
both 2D and 3D cases) achieves an order of magnitude speedup over standard
computational geometry libraries.Comment: 11 page
Decomposition Methods for Large Scale LP Decoding
When binary linear error-correcting codes are used over symmetric channels, a
relaxed version of the maximum likelihood decoding problem can be stated as a
linear program (LP). This LP decoder can be used to decode error-correcting
codes at bit-error-rates comparable to state-of-the-art belief propagation (BP)
decoders, but with significantly stronger theoretical guarantees. However, LP
decoding when implemented with standard LP solvers does not easily scale to the
block lengths of modern error correcting codes. In this paper we draw on
decomposition methods from optimization theory, specifically the Alternating
Directions Method of Multipliers (ADMM), to develop efficient distributed
algorithms for LP decoding.
The key enabling technical result is a "two-slice" characterization of the
geometry of the parity polytope, which is the convex hull of all codewords of a
single parity check code. This new characterization simplifies the
representation of points in the polytope. Using this simplification, we develop
an efficient algorithm for Euclidean norm projection onto the parity polytope.
This projection is required by ADMM and allows us to use LP decoding, with all
its theoretical guarantees, to decode large-scale error correcting codes
efficiently.
We present numerical results for LDPC codes of lengths more than 1000. The
waterfall region of LP decoding is seen to initiate at a slightly higher
signal-to-noise ratio than for sum-product BP, however an error floor is not
observed for LP decoding, which is not the case for BP. Our implementation of
LP decoding using ADMM executes as fast as our baseline sum-product BP decoder,
is fully parallelizable, and can be seen to implement a type of message-passing
with a particularly simple schedule.Comment: 35 pages, 11 figures. An early version of this work appeared at the
49th Annual Allerton Conference, September 2011. This version to appear in
IEEE Transactions on Information Theor
Convex Hulls: Surface Mapping onto a Sphere
Writing an uncomplicated, robust, and scalable three-dimensional convex hull
algorithm is challenging and problematic. This includes, coplanar and collinear
issues, numerical accuracy, performance, and complexity trade-offs. While there
are a number of methods available for finding the convex hull based on
geometric calculations, such as, the distance between points, but do not
address the technical challenges when implementing a usable solution (e.g.,
numerical issues and degenerate cloud points). We explain some common algorithm
pitfalls and engineering modifications to overcome and solve these limitations.
We present a novel iterative method using support mapping and surface
projection to create an uncomplicated and robust 2d and 3d convex hull
algorithm
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